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Cost Estimating Using a New Learning Curve Theory for Non-Constant Production Rates

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Cost Estimating Using a New Learning Curve Theory for Non-Constant Production Rates forecasting Article Cost Estimating Using a New Learning Curve Theory for Non-Constant Production Rates Dakotah Hogan 1, John Elshaw 2,*, Clay Koschnick 2, Jonathan Ritschel 2, Adedeji Badiru 3 and Shawn Valentine 4 1 Air Force Cost Analysis Agency, Deputy Assistant Secretary for Cost and Economics, Joint Base Andrews, MD 20762, USA; [email protected] 2 Department of Systems Engineering & Management, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA; clay.koschnick@afit.edu (C.K.); jonathan.ritschel@afit.edu (J.R.) 3 Graduate School of Engineering and Management, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA; adedeji.badiru@afit.edu 4 Estimating Research & Technology Advising Branch, Cost and Economics Division, Air Force Lifecycle Management Center, Wright-Patterson AFB, OH 45433, USA; [email protected] * Correspondence: john.elshaw@afit.edu; Tel.: +1-937-255-3636 (ext. 4650) Received: 27 August 2020; Accepted: 9 October 2020; Published: 16 October 2020 Abstract: Traditional learning curve theory assumes a constant learning rate regardless of the number of units produced. However, a collection of theoretical and empirical evidence indicates that learning rates decrease as more units are produced in some cases. These diminishing learning rates cause traditional learning curves to underestimate required resources, potentially resulting in cost overruns. A diminishing learning rate model, namely Boone’s learning curve, was recently developed to model this phenomenon. This research confirms that Boone’s learning curve systematically reduced error in modeling observed learning curves using production data from 169 Department of Defense end-items. However, high amounts of variability in error reduction precluded concluding the degree to which Boone’s learning curve reduced error on average. This research further justifies the necessity of a diminishing learning rate forecasting model and assesses a potential solution to model diminishing learning rates. Keywords: learning curve; forecasting; production cost; cost estimating 1. Introduction The U.S. Government Accountability Office (GAO) critiqued the cost and schedule performance of the Department of Defense (DoD)’s $1.7 trillion portfolio of 86 major weapons systems in their 2018 “Weapons System Annual Assessment.” The GAO cited realistic cost estimates as a reason for the relatively low cost growth of the portfolio in comparison to earlier portfolios [1]. Congress and its oversight committees maintain a watchful eye on the DoD’s complex and expensive weapons system portfolio. Inefficient programs are scrutinized and may be terminated if inefficiencies persist. Funding of inefficient programs will also lead to the underfunding of other programs. In the public sector, these terminated and underfunded programs may result in capability gaps that negatively impact our nation’s defense. In the private sector, the inefficient use of resources often spells failure for a company. A key to the efficient use of resources is accurately estimating the resources required to produce an end-item. Learning curves are a popular method of forecasting required resources as they predict end-item costs using the item’s sequential unit number in the production line. Learning curves are especially useful when estimating the required resources for complex products. The most popular learning curve models used in the government sector are over 80 years old and may be outdated in Forecasting 2020, 2, 429–451; doi:10.3390/forecast2040023 www.mdpi.com/journal/forecasting Forecasting 2020, 2 430 today’s technology-rich production environment. Additionally, researchers have demonstrated both theoretically and empirically that the effects of learning slow or cease over time [2–4]. A new model, named Boone’s learning curve, has been recently proposed to account for diminishing rates of learning as more units are produced [5]. The purpose of this research is to survey the need for alternative learning curve models and further examine how Boone’s learning curve performs in comparison to the traditional learning curve theories in predicting required resources. This research uses a large number of diverse production items to compare Boone’s model to the traditional theories of Wright and Crawford. While many different learning curve models exist (i.e., DeJong, Stanford B, Sigmoid, etc.), some of these others may not be as accurate in cases where the learning rate decreases over time. The next section is a review of the learning curve literature relevant to diminishing learning rates, followed by a description of our methodology and analysis to compare Boone’s learning curve to traditional models. We conclude the paper discussing managerial implications and limitations followed by recommendations for the way forward. 2. Literature Review and Background The two learning curve models cited by the GAO Cost Estimating and Assessment Guide (2009) are Wright’s cumulative average learning curve theory developed in 1936 and Crawford’s unit learning curve theory developed in 1947. Although both learning curve theories use the same general equation, the theories have contrasting variable definitions. Wright’s learning curve is shown in Equation (1): Y = Axb (1) where Y is the cumulative average cost of the first x units, A is the theoretical cost to produce the first unit, x is the cumulative number of units produced, and b is the natural logarithm of the learning curve slope (LCS) divided by the natural logarithm of two. Note, the LCS is the complement of the percent decrease in cost as the number of units produced doubles. For example, with a learning curve slope of 80% and a first unit cost of 100 labor hours, the average cost of the first two units would be 80 labor hours, or 60 labor hours for the second unit. Regardless of the number of units produced, there is a constant decrease in labor costs with each doubling of units due to the constant learning rate. Several years following the creation of Wright’s cumulative average learning curve theory, J.R. Crawford formulated the unit learning curve theory. Crawford’s theory deviates from Wright’s by assuming that the individual unit cost (as opposed the cumulative average unit cost) decreases by a constant percentage as the number of units produced doubles. Crawford’s model is shown in Equation (2): Y = Axb (2) where Y is the individual cost of unit x, A is the theoretical cost of the first unit, x is the unit number of the unit cost being forecasted, and b is the natural logarithm of the LCS divided by the natural logarithm of two. For example, with a learning curve slope of 80% and a first unit cost of 100 labor hours, the cost of the second unit would be 80 labor hours. Note, Crawford’s unit theory is the similar to Wright’s in function form; but the difference arises in the variable interpretation lead to a different forecast. Figure1 below shows a comparison between Wright’s and Crawford’s theories using the two numerical examples provided. Cumulative average theory and unit theory will produce different predicted costs provided the same set of data despite all predicted costs being normalized to unit costs. Figure1 demonstrates this point where unit theory was used to generate data using a first unit cost of 100 and a learning curve slope of 90%. The original unit theory data was converted to cumulative averages in order to estimate cumulative average theory learning curve parameters. Forecasting 2020, 2 FOR PEER REVIEW 3 parameters were then used to predict cumulative average costs. These predicted costs were then converted to unit costs. This conversion allows for the cumulative average predictions to be directly compared to the original Unit Theory generated data. As shown in Figure 1, the cumulative average learning curve predictions first overestimate, then underestimate, and ultimately overestimate the Forecastinggenerated2020 unit, 2 theory data for all remaining units. Together, Wright’s and Crawford’s theories form431 the basis of the traditional learning curve theory. Figure 1. Wright’sWright’s Cumulative Average TheoryTheory vs. Crawford’s Unit Theory. CumulativeOne assumption average of these theory traditional learning learning curve parameters. curve theories Cumulative is that they average only apply theory to estimatedprocesses athat learning may benefit curve from slope learning. of 93% andTypically, a first these unit costcosts of are 101.24. only a Thesesubset Cumulativeof total program Average costs; Theory hence parametersappropriate werecosts must then usedbe considered to predict when cumulative applying average learning costs. curve These theory predicted to yield viable costs wereparameter then convertedestimates. toIn unita complex costs. 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